Trigonometric Identities


Uploaded by khanacademy on 27.10.2007

Transcript:

Trigonometric Identities


Welcome back
I am now going to do a series of videos on the trigonometric identities
So let's just review what we already know about the trig functions
so if I just write soh-cah-toa
soh-cah-toa
that tells us... and now that we've actually extended this with the unit circle definition... but if you watch those videos
you'll realize that the unit circle definition directly uses soh-cah-toa
so we'll just stick with soh-cah-toa
because I think it will help make some of what we are about to do seem a little bit more straightforward
and we'll kind of verge on the unit circle definition anyway
so we know that sine of theta
sine of theta is equal to Opposite over hypotenuse, right?
so, cosine of theta is equal to Adjacent over hypotenuse
and then the tangent of theta is equal to opposite over adjacent
so let's draw that out on a right triangle
we can do this with the unit circle as well
and it would make sense
let's see if we can find a relationship between sine, cosine and tangent
there's my right triangle
let's call this theta
this is the hypotenuse h
this is the opposite side, right? the opposite of theta
this is theta right here
this is the adjacent side, right?
well what do we know about the relationship between the opposite, the adjacent side and the hypotenuse?
what does the Pythagorean theorem tell us?
oh yeah, the opposite... this side squared plus this side squared is equal to the hypotenuse squared
so we can write that down
'a' squared plus 'o' squared is equal to the hypotenuse squared, right?
and then, this is just an equation, so if we want to we could divide both sides of this equation by h... by h squared
so what do we get?
we get 'a' squared over 'h' squared.. plus.. 'o' squared over 'h' squared is equal to one, right?
and then I could rewrite that as
('a' over 'h') squared plus ('o' over 'h') squared is equal to one
now do these look at all familiar?
hmm... what we have over here right?
this is 'a' over 'h'
and this is 'o' over 'h'
so we can just substitute
so this is just cosine of theta
cosine of theta squared and this is how you write cosine squared...
you could put parentheses round the whole thing and square it but this is just the notation people use
plus opposite over adjacent hypotenuse squared, so that's sine theta squared equal to one
so that's our first trig identity
so if you know the sine of theta it's very easy to figure out the cosine of theta, right?
you can just solve this equation
right if I know that.. I don't know, let's say I know that the sine of theta
let's say that I know the sine of theta is let's say one half, right?
then what is the cosine of theta?
cosine of theta is what?
well I know the sine of theta is one half, right so
cosine squared of theta plus
sine of theta is one half so
one half squared
is equal to one, right?
cosine squared theta plus one fourth is equal to one
so we have cosine squared theta is equal to three fourths
or cosine of theta would be the square root of this, we take the square root of both sides
the square root of three over two
and you probably remember that from our whole presentation on the 30-60-90 triangle
so I just wanted to show you a use of this trig identity
it's usually written sine squared plus cosine squared is equal to one
so let's just extend that one a little bit
let's play with the ratios and see what else we can... what other... identities... I guess, we can discover
oh whoops
clear "image invert colors"
so we know that sine squared theta plus cosine squared theta is equal to one
one thing we could do is divide both sides of this equation by cosine squared of theta
and let's just see what happens when we do that
so if we say... cosine squared theta... you have to distribute across... both terms
cosine squared theta... and then cosine squared of theta
well what's sine squared theta over cosine squared theta?
that's the same thing as ( sine of theta over cosine of theta ) squared
plus this is one
is equal to ( one over cosine theta ) squared
right, I just , I mean, one squared is one, I just rewrote it
so sine over cosine theta, I think we learned that already
that's just the tangent of theta
and in case you haven't learned that already, think about it this way
sine is opposite over the hypotenuse, right?
so that's opposite over hypotenuse
and then cosine is adjacent over hypotenuse
so adjacent over hypotenuse
and that equals opposite over hypotenuse times hypotenuse over adjacent
right, just dividing by a fraction is the same thing as multiplying by its reciprocal... that's all I did
and that equals opposite over adjacent, right?
so that equals opposite over adjacent
that just says sine of theta over cosine of theta is equal to tangent of theta
so sine squared theta over cosine squared theta is tan squared theta
and then plus one is ( one over cosine theta ) squared
and now I'm going to introduce a new trig ratio and it is really just one over cosine theta
one over cosine theta... I'm going to summarize this at the end just so that it's not too confusing
is actually the secant of theta
and it's just another ratio
the secant of theta instead of being the adjacent over the hypotenuse would be the hypotenuse over the adjacent, right?
it's just one over cosine theta, nothing famous, uh, nothing fancy here
so secant of theta
so that equals secant squared of theta
I know it can be a little overwhelming initially just because I'm throwing out all of these new terms
secant is one over cosine theta
but once you just play around with these enough and get familiar with the terms it'll make sense it'll be more natural to you
so this could be... you could view this as another trig identity
and actually in case... I don't even remember if I've taught it already you could view this as a trig identity... though that's almost definitional
and then of course you can, in case I haven't done it already
you now know that sine of theta over cosine of theta is equal to tangent of theta
and that's right here... I guess you could say the proof of it
so let me keep introducing you to more things
if this is really daunting maybe you can just rewatch it and hopefully it will make some sense
ah... let me see... clear.. image
so what have we learned so far?
we've learned that sine squared theta plus cosine squared theta is equal to one
we learned that sine of theta over cosine of theta is equal to tangent of theta
we learned that the tangent squared of theta plus one is equal to the secant of theta squared...
let me write this definition down...
oops... is equal to the secant squared of theta... sorry
and the secant of theta is just one over cosine of theta
and this is really something that you should just memorise
secant is one over cosine
and if you're wondering what one over sine is
one over sine of theta
it's the co-secant
the abbreviation is csc of theta
and if you're wondering what one over the tangent is
this is the co-tangent
and you just might want to memorize these
and this often confuses me... that one over the cosine is the secant but one over the sine is the cosecant
so it's almost the opposite, right?
one over the sine has a 'co' in it while one over the cosine doesn't have the 'co' in it
so, that might help you, uh, remember things
so I think that's all I have time for now
in the next presentation I'm going to introduce you to a couple more trig identities
see you soon